On the smoothness of the equizonal ovaloids

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• 作者：Vincent Coll (1)
  Jeff Dodd (2)
  Michael Harrison (3)
  
• 关键词：53A07 ; 26A46 ; Equizonal ovaloids ; equal area zones property ; hypersurfaces of revolution ; smoothness ; absolutely monotonic functions ; completely monotonic functions
• 刊名：Journal of Geometry
• 出版年：2012
• 出版时间：December 2012
• 年：2012
• 卷：103
• 期：3
• 页码：409-416
• 参考文献：1. Bernstein S.: Sur la définition et les propriétés des fonctions analytiques d’une variable réelle. Math. Ann. 75, 449-68 (1914) CrossRef
  2. Blaschke W.: Vorlesungen über Differentialgeometrie I. Springer, Berlin (1921)
  3. Dodd J., Coll V.: Generalizing the equal area zones property of the sphere. J. Geom. 90, 47-5 (2008) CrossRef
  4. Richmond B., Richmond T.: The equal area zones property. Am. Math. Mon. 100, 475-77 (1993) CrossRef
  5. Stamm, O.: / Umkehrung eines Satzes von Archimedes über die Kugel. Abh. Math. Sem. Univ. Hamburg 17, 112-32 (1951); reviewed in MathSciNet: MR0041467
  6. Widder D.: The Laplace Transform. Princeton University Press, Princeton (1941)
  
• 作者单位：Vincent Coll (1)
  Jeff Dodd (2)
  Michael Harrison (3)
  
  1. Department of Mathematics, Lehigh University, Christmas-Saucon Hall, 14 E. Packer Avenue, Bethlehem, PA, 18015, USA
  2. Mathematical, Computing, and Information Sciences Department, Jacksonville State University, Jacksonville, AL, 36265, USA
  3. The Pennsylvania State University, 109A McAllister Building, University Park, PA, 16802, USA
  
• ISSN：1420-8997

文摘

We show that the n-dimensional equizonal ovaloids are analytic when n is even and are of exactly C n-1 smoothness when n is odd. This substantially improves the previously published result on the smoothness of the even-dimensional equizonal ovaloids and slightly corrects the previously published statement regarding the smoothness of the odd-dimensional equizonal ovaloids. Our methods should be generally useful in determining the degree of smoothness of surfaces and hypersurfaces of revolution generated by piecewise-defined profile curves. In particular, they include a novel and elegant application of Bernstein’s theory of absolutely monotonic functions.